Which part of the plugin to change so the updater won’t register the next version?

You should not modify Themes or Plugins that you don’t control. Instead, use one of two options:

1. Patch the Theme/Plugin, and submit the patch to the developer
2. Fork the Theme/Plugin

It sounds like you’ve opted for option #2. In that case, I would recommend changing both of the following:

1. Update the entire header phpDoc refs to your forked version of the Plugin
2. Change the directory name from /wp-content/plugins/{plugin-name}, to /wp-content/plugins/{forked-plugin-name}

I believe the bare-minimum change needed is changing the Plugin name phpDoc, and the Plugin’s directory name.